Friedrichs extension explained

In functional analysis, the Friedrichs extension is a canonical self-adjoint extension of a non-negative densely defined symmetric operator. It is named after the mathematician Kurt Friedrichs. This extension is particularly useful in situations where an operator may fail to be essentially self-adjoint or whose essential self-adjointness is difficult to show.

An operator T is non-negative if

\langle \xi \mid T \xi \rangle \geq 0 \xi \in \operatorname{dom} T

Examples

Example. Multiplication by a non-negative function on an L² space is a non-negative self-adjoint operator.

Example. Let U be an open set in Rⁿ. On L²(U) we consider differential operators of the form

[T \phi](x) = - \sum _{i,j} \partial _{x _{i}} \{a _{i} (x) \partial _{x _{j}} \phi(x)\} x \in U, \phi \in \operatorname{C} ^{inf ty(U),} _{c}

where the functions a_{i j} are infinitely differentiable real-valued functions on U. We consider T acting on the dense subspace of infinitely differentiable complex-valued functions of compact support, in symbols

\operatorname{C} ^{inf ty(U)} _{c} \subseteq L ^{2(U).}

If for each x ∈ U the n × n matrix

\begin{bmatrix} a _{1} (x) & a _{1} (x) & … & a _{1} (x) \ a _{2} (x) & a _{2} (x) & … & a _{2} (x) \ \vdots & \vdots & \ddots & \vdots \ a _{n} (x) & a _{n} (x) & … & a _{n} (x) \end{bmatrix}

is non-negative semi-definite, then T is a non-negative operator. This means (a) that the matrix is hermitian and

\sum _{i,} a _{i} (x) c _{i} \overline{c _{j}} \geq 0

for every choice of complex numbers c₁, ..., c_n. This is proved using integration by parts.

These operators are elliptic although in general elliptic operators may not be non-negative. They are however bounded from below.

Definition of Friedrichs extension

The definition of the Friedrichs extension is based on the theory of closed positive forms on Hilbert spaces. If T is non-negative, then

\operatorname{Q}(\xi, η ) = \langle \xi \mid T η \rangle + \langle \xi \mid η \rangle

is a sesquilinear form on dom T and

\operatorname{Q}(\xi, \xi) = \langle \xi \mid T \xi\rangle + \langle \xi \mid \xi \rangle \geq \|\xi\| ^{2.}

Thus Q defines an inner product on dom T. Let H₁ be the completion of dom T with respect to Q. H₁ is an abstractly defined space; for instance its elements can be represented as equivalence classes of Cauchy sequences of elements of dom T. It is not obvious that all elements in H₁ can be identified with elements of H. However, the following can be proved:

The canonical inclusion

\operatorname{dom} T → H

extends to an injective continuous map H₁ → H. We regard H₁ as a subspace of H.

Define an operator A by

\operatorname{dom} A = \{\xi \in H _{1:} \phi _{\xi:} η \mapsto \operatorname{Q}(\xi, η ) isboundedlinear. \}

In the above formula, bounded is relative to the topology on H₁ inherited from H. By the Riesz representation theorem applied to the linear functional φ_ξ extended to H, there is a unique A ξ ∈ H such that

\operatorname{Q}(\xi, η ) = \langle A \xi \mid η \rangle η \in H _{1}

Theorem. A is a non-negative self-adjoint operator such that T₁=A - I extends T.

T₁ is the Friedrichs extension of T.

Another way to obtain this extension is as follows. Let :

L:H _{1 →} H be the bounded inclusion operator. The inclusion is a bounded injective with dense image. Hence LL ^{*:H →} H is a bounded injective operator with dense image, where L ^{*} is the adjoint of L as an operator between abstract Hilbert spaces. Therefore the operator A: = (LL ^{*)} ^{-1} is a non-negative self-adjoint operator whose domain is the image of LL ^{*} . Then A - I extends T.

Krein's theorem on non-negative self-adjoint extensions

M. G. Krein has given an elegant characterization of all non-negative self-adjoint extensions of a non-negative symmetric operator T.

If T, S are non-negative self-adjoint operators, write

T \leq S

if, and only if,

\operatorname{dom}(S ^{1/2} ) \subseteq \operatorname{dom}(T ^{1/2} ) \langle T ^{1/2} \xi \mid T ^{1/2} \xi \rangle \leq \langle S ^{1/2} \xi \mid S ^{1/2} \xi \rangle \forall \xi \in \operatorname{dom}(S ^{1/2} )

Theorem. There are unique self-adjoint extensions T_min and T_max of any non-negative symmetric operator T such that

T _{min} \leq T _{max} ,

and every non-negative self-adjoint extension S of T is between T_min and T_max, i.e.

T _{min} \leq S \leq T _{max} .

See also

• Energetic extension
• Extensions of symmetric operators

References

• N. I. Akhiezer and I. M. Glazman, Theory of Linear Operators in Hilbert Space, Pitman, 1981.